3.4 Properties Of Double Integrals

1.: \(\displaystyle {\iint \limits _D \big (f(x,y) + g(x,y)\big )\hspace {0.1cm}dA = \iint \limits _D f(x,y)\hspace {0.1cm}dA + \iint \limits _D g(x,y)\hspace {0.1cm}dA}\)
2.: \(\displaystyle { \iint \limits _D C\hspace {0.1cm} f(x,y)\hspace {0.1cm}dA = C \iint \limits _D f(x,y)\hspace {0.1cm}dA }\)
3.: If \(f(x,y)\geq g(x,y)\) for all \((x,y)\in D\) then \(\displaystyle {\iint \limits _D f(x,y)\hspace {0.1cm}dA \geq \iint \limits _D g(x,y)\hspace {0.1cm}dA }\).
4.: If \(D = D_1\cup D_2\) where \(D_1\) and \(D_2\) do not overlap except perhaps on the boundary \[\iint \limits _D f(x,y)\hspace {0.1cm}dA = \iint \limits _{D_1} f(x,y)\hspace {0.1cm}dA + \iint \limits _{D_2} f(x,y)\hspace {0.1cm}dA \]
5.: If we integrate the constant function \(f(x,y) = 1\) over a region \(D\) we get the area of \(D\) \[\iint \limits _D 1 \hspace {0.1cm}dA = A(D)\]
6.: If \(m\leq f(x,y) \leq M\) for all \((x,y) \in D\) then \[m\hspace {0.1cm}A(D)\leq \iint \limits _D f(x,y)\hspace {0.1cm}dA \leq M\hspace {0.1cm} A(D)\]

Example 3.4.1.

Use property 6 to estimate the integral \(\displaystyle {\iint \limits _De^{\displaystyle {\sin x \cos y}}\hspace {0.1cm} dA}\) where \(dA\) is the disc with centre origin and radius 2.

Solution. \begin {align*} \text {Since}\hspace {1cm} -1 & \leq \sin x \leq 1\\ -1 & \leq \cos y \leq 1 \hspace {1cm} \implies -1 \leq \sin x \cos y\leq 1 \end {align*}

\begin {align*} e^{-1} & \leq e^{\displaystyle {\sin x \cos y}} \leq e^1\\ \frac {1}{e} & \leq e^{\displaystyle {\sin x \cos y}} \leq e\\ \end {align*}

Using \(m = \dfrac {1}{e}\) and \(M = e\)

Area of disc \(= \pi (2)^2 = 4\pi \)

\[\therefore \hspace {0.4cm} \frac {4\pi }{e}\leq \iint \limits _D e^{\displaystyle {\sin x \cos y}}\hspace {0.1cm} dA \leq 4\pi e\]

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